Abstract
The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, we provide various approximation results concerning the classical Korovkin theorem via Fibonacci type statistical convergence.
Highlights
1.1 Densities and statistical convergence Let A be a subset of positive integers
4 Conclusion One of the most known and interesting number sequences is the Fibonacci sequence, and it still continues to be of interest to mathematicians because this sequence is an important and useful tool to expand the mathematical horizon for many mathematicians
Statistical convergence has recently become an area of active research
Summary
1.1 Densities and statistical convergence Let A be a subset of positive integers. We consider the interval [ , n] and select an integer in this interval, randomly. The ratio of the number of elements of A in [ , n] to the total number of elements in [ , n] belongs to A, probably. For n → ∞, if this probability exists, that is, this probability tends to some limit, this limit is used as the asymptotic density of the set A. Let us mention that the asymptotic density is a kind of probability of choosing a number from the set A. The set of positive integers will be denoted by Z+. If the symmetric difference A B is finite, we can say A is asymptotically equal to B and denote A ∼ B. Freedman and Sember introduced the concept of a lower asymptotic density and defined the concept of convergence in density, in [ ]
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